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30
A Conjectured Combinatorial Formula for the Hilbert . . .
, 2004
"... We introduce a conjectured way of expressing the Hilbert series of diagonal harmonics as a weighted sum over parking functions. Our conjecture is based on a pair of statistics for the q,tCatalan sequence discovered by M. Haiman and proven by the first author and A. Garsia (Proc. Nat. Acad. Sci. 98 ..."
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Cited by 25 (13 self)
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We introduce a conjectured way of expressing the Hilbert series of diagonal harmonics as a weighted sum over parking functions. Our conjecture is based on a pair of statistics for the q,tCatalan sequence discovered by M. Haiman and proven by the first author and A. Garsia (Proc. Nat. Acad. Sci. 98 (2001), 43134316). We show how our q,tparking function formula for the Hilbert series can be expressed more compactly as a sum over permutations. We also derive two equivalent forms of our conjecture, one of which is based on the original pair of statistics for the q,tCatalan introduced by the first author and the other of which is expressed in terms of rooted, labelled trees.
An Introduction to Hyperplane Arrangements
 Lecture notes, IAS/Park City Mathematics Institute
, 2004
"... ..."
Parking Functions, Empirical Processes, and the Width of Rooted Labeled Trees
"... This paper provides tight bounds for the moments of the width of rooted labeled trees with n nodes, answering an open question of Odlyzko and Wilf (1987). To this aim, we use one of the many onetoone correspondences between trees and parking functions, and also a precise coupling between parking f ..."
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Cited by 18 (6 self)
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This paper provides tight bounds for the moments of the width of rooted labeled trees with n nodes, answering an open question of Odlyzko and Wilf (1987). To this aim, we use one of the many onetoone correspondences between trees and parking functions, and also a precise coupling between parking functions and the empirical processes of mathematical statistics. Our result turns out to be a consequence of the strong convergence of empirical processes to the Brownian bridge (Komlos, Major and Tusnady, 1975).
k, m)Catalan numbers and hook length polynomials for trees
"... Abstract: In this paper we define and study the (k,m)Catalan number Ck,m(n) = 1 mn+1 n which is a generalization of the Catalan number C(n) = 1 2n n+1 n, and give two combinatorial interpretations for this number: (k,m)ary trees with n crucial vertices and (mn+1)×k “parking tables”. Using these ..."
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Cited by 13 (1 self)
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Abstract: In this paper we define and study the (k,m)Catalan number Ck,m(n) = 1 mn+1 n which is a generalization of the Catalan number C(n) = 1 2n n+1 n, and give two combinatorial interpretations for this number: (k,m)ary trees with n crucial vertices and (mn+1)×k “parking tables”. Using these interpretations we give recurrence formulas for Ck,m(n) which can be
A simple bijection for the regions of the Shi arrangement of hyperplanes
, 1997
"... The Shi arrangement Sn is the arrangement of affine hyperplanes in R n of the form xi−xj = 0 or 1, for 1 ≤ i < j ≤ n. It dissects R n into (n+1) n−1 regions, as was first proved by Shi. We give a simple bijective proof of this result. Our bijection generalizes easily to any subarrangement of Sn c ..."
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Cited by 11 (2 self)
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The Shi arrangement Sn is the arrangement of affine hyperplanes in R n of the form xi−xj = 0 or 1, for 1 ≤ i < j ≤ n. It dissects R n into (n+1) n−1 regions, as was first proved by Shi. We give a simple bijective proof of this result. Our bijection generalizes easily to any subarrangement of Sn containing the hyperplanes xi − xj = 0 and to the extended Shi arrangements.
On the Sandpile Group of a Graph
 European Journal of Combinatorics
, 2000
"... We show how to express the sandpile model, introduced in theoretical physics, using the vocabulary of combinatorial theory. The group of recurrent configurations in the sandpile model, introduced by D. Dhar ([6]), may be considered as a finite abelian group associated with any graph G; we call it th ..."
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Cited by 10 (1 self)
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We show how to express the sandpile model, introduced in theoretical physics, using the vocabulary of combinatorial theory. The group of recurrent configurations in the sandpile model, introduced by D. Dhar ([6]), may be considered as a finite abelian group associated with any graph G; we call it the sandpile group of G. The structure of the sandpile group is determined for some families of graphs.
Conjectured combinatorial models for the Hilbert series of generalized diagonal harmonics modules
 Electron.J.Combin.11 (2004), R68; 64
"... Haglund and Loehr previously conjectured two equivalent combinatorial formulas for the Hilbert series of the GarsiaHaiman diagonal harmonics modules. These formulas involve weighted sums of labelled Dyck paths (or parking functions) relative to suitable statistics. This article introduces a third c ..."
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Cited by 6 (4 self)
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Haglund and Loehr previously conjectured two equivalent combinatorial formulas for the Hilbert series of the GarsiaHaiman diagonal harmonics modules. These formulas involve weighted sums of labelled Dyck paths (or parking functions) relative to suitable statistics. This article introduces a third combinatorial formula that is shown to be equivalent to the first two. We show that the four statistics on labelled Dyck paths appearing in these formulas all have the same univariate distribution, which settles an earlier question of Haglund and Loehr. We then introduce analogous statistics on other collections of labelled lattice paths contained in trapezoids. We obtain a fermionic formula for the generating function for these statistics. We give bijective proofs of the equivalence of several forms of this generating function. These bijections imply that all the new statistics have the same univariate distribution. Using these new statistics, we conjecture combinatorial formulas for the Hilbert series of certain generalizations of the diagonal harmonics modules. 1
Counting Defective Parking Functions
, 803
"... Suppose that m drivers each choose a preferred parking space in a linear car park with n spaces. Each driver goes to the chosen space and parks there if it is free, and otherwise takes the first available space with a larger number (if any). If all drivers park successfully, the sequence of choices ..."
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Cited by 3 (0 self)
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Suppose that m drivers each choose a preferred parking space in a linear car park with n spaces. Each driver goes to the chosen space and parks there if it is free, and otherwise takes the first available space with a larger number (if any). If all drivers park successfully, the sequence of choices is called a parking function. In general, if k drivers fail to park, we have a defective parking function of defect k. Let cp(n,m,k) be the number of such functions. In this paper, we establish a recurrence relation for the numbers cp(n,m,k), and express this as an equation for a threevariable generating function. We solve this equation using the kernel method, and extract the coefficients explicitly: it turns out that the cumulative totals are partial sums in Abel’s binomial identity. Finally, we compute the asymptotics of cp(n,m,k). In particular, for the case m = n, if choices are made independently at random, the limiting distribution of the defect (the number of drivers who fail to park), scaled by the square root of n, is the Rayleigh distribution. On the other hand, in the case m = ω(n), the probability that all spaces are occupied tends asymptotically to one. 1
Parking functions and vertex operators
"... Abstract. We introduce several associative algebras and series of vector spaces associated to these algebras. Using lattice vertex operators, we obtain dimension and character formulae for these spaces. In particular, we a series of representations of symmetric groups which turn out to be isomorphic ..."
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Cited by 2 (1 self)
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Abstract. We introduce several associative algebras and series of vector spaces associated to these algebras. Using lattice vertex operators, we obtain dimension and character formulae for these spaces. In particular, we a series of representations of symmetric groups which turn out to be isomorphic to parking function modules. We also construct series of vector spaces whose dimensions are Catalan numbers and Fuss–Catalan numbers respectively. Conjecturally, these spaces are related to spaces of global sections of vector bundles on (zero fibres of) Hilbert schemes and representations of rational Cherednik algebras. 1.